Hey Sara, these things happen. And typically when we are on a deadline or when things just aren't going so well and everything was perfect and dainty until great huge doggy footprints messed everything up....
Many people call a top a "flimsy" because that is what it is, very flimsy and fragile. Unless there are actual bodily fluids like pee or barf (I'm looking at you, cat), I would gently blot as much up as you can while damp and then let it dry and brush off as much as possible.
Once you get the top quilted together and well stabilized, that is the time to wash even if you don't usually wash your quilts.
If you decide you absolutely positively have to wash things before quilting down, put small amounts in lingerie bags or pillow cases. Take out while damp and do what you can to straighten them out, maybe air fluff for a bit. Then I would take them damp and layout a towel on my ironing board and press them (being careful not to stretch) flat and dry.